Problem 24 - Chemistry The Science in Context

Question

Super fluids The 1996 Nobel Prize in Physics was awarded to Douglas Osheroff, Robert Richardson, and David Lee for discovering super-fluidity (apparently frictionless flow) in "He. When \(^{3} \mathrm{He}\) is cooled to \(2.7 \mathrm{mK}\), the liquid settles into an ordered super-fluid state. What is the predicted sign of the entropy change for the conversion of liquid ' He into its super-fluid state?

Step-by-Step Solution

Answer

The sign of entropy change for the conversion of liquid helium into its superfluid state is negative, indicating a decrease in entropy. This is because the system becomes more ordered during the transition and loses heat to its surroundings.

Step 1: Understand entropy change

Entropy change, denoted by \(\Delta S\), is a measure of the change in the number of microstates in a system. If a system becomes more ordered, as it does when it transitions from a liquid state to a superfluid state, the number of possible microstates the particles can occupy decreases, resulting in a negative entropy change. Since entropy change is linked to the heat transfer and temperature, we can also check the sign using the formula: $$\Delta S = \frac{q}{T}$$ where \(q\) represents heat transfer and \(T\) is the temperature in Kelvin.

Step 2: Analyze heat transfer during the transition of liquid helium to superfluid

As helium is cooled to \(2.7\,\text{mK}\) to undergo the phase transition, it means that the system is losing heat to the surroundings, hence \(q\) is negative.

Step 3: Analyze temperature sign during the process

The given temperature is \(2.7\,\text{mK}\). Since a temperature can never be negative, the value of \(T\) is positive.

Step 4: Determine the sign of entropy change using the formula

Using the formula \(\Delta S = \frac{q}{T}\), since \(q\) is negative and \(T\) is positive, the sign of \(\Delta S\) will be negative, indicating a decrease in entropy. So, the predicted sign of the entropy change for the conversion of liquid helium into its superfluid state is negative.

Key Concepts

Entropy Change

When we talk about entropy change in the context of superfluidity, we are referring to how "chaotic" or "ordered" the system's particles become. Entropy, in simple terms, is a measure of disorder. If a system becomes more organized—like when liquid helium transitions into a superfluid—it means there's less randomness or fewer microstates available.
This implies a decrease in entropy. In mathematical terms, the entropy change represented by \(\Delta S = \frac{q}{T}\)can be used to determine the behavior of a system during a transition.
In this formula:

• \(q\) represents the amount of heat transfer—whether the system absorbs or releases heat.
• \(T\) is the temperature in Kelvin, ensuring it remains positive.

When we go from a liquid to a superfluid state, the heat is released to the environment (\(q\) is negative), causing \(\Delta S\)to be negative, as the particles settle into a more orderly state.

Phase Transition

Phase transitions happen when a substance changes from one state of matter to another, like liquid to gas, or in this case, liquid helium-3 turning into a superfluid.
Superfluidity is a remarkable state where helium flows without any internal friction, resulting in incredibly unique properties. For such a phase change to occur, conditions must be just right—typically at extremely low temperatures.
For helium-3, this occurs around \(2.7 \, \text{mK}\).
During this transition:

• Energy in the form of heat is released as the system becomes more ordered.
• The molecular motion changes, allowing the particles to condense into a super-ordered phase.

While most transitions occur with heat absorption, the superfluid transition involves heat release, aligning with the idea of decreasing entropy.

Helium-3

Helium-3 \(\left(^{3}\mathrm{He}\right)\) is a rare isotope of helium that acts quite differently compared to the more common helium-4. One of the most interesting behaviors of helium-3 is its ability to become a superfluid very close to absolute zero. This superfluid state arises at approximately \(2.7 \mathrm{mK}\).
Its uniqueness comes from its atomic structure:

• Helium-3 atoms are "fermions," meaning they obey Fermi-Dirac statistics.
• At low temperatures, they pair up to form "bosons," which can occupy the same quantum state—this is crucial for superfluidity.

Understanding helium-3's properties helps in studying quantum mechanics and low-temperature physics. Its transition into a superfluid state at such low temperatures has fascinating implications for both theoretical and applied sciences, offering insights into the quantum world and superfluid behavior.